Computational and algorithmic linear algebra by Murty K.G.

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By Murty K.G.

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From calculus we know that a necessary condition for (¯a0 , . . , a ¯n ) to minimize L2 (a0 , . . , an ) is that it must satisfy the equations obtained by setting the partial derivatives of L2 (a0 , . . , an ) WRT a0 , . . , an equal to 0. ∂L2 (a0 , . . , an ) = 0 j = 0 to n. ∂aj This system is known as the system of normal equations. When the model function f (t) satisfies certain conditions, the system of normal equations is a system of linear equations that can be solved by the methods to be discussed later on in this chapter.

2 Example applications 39 (ii): CH4 is the formula for methane which is the main component in natural gas. Balance the chemical reaction in which methane combines with oxygen to produce carbon dioxide and water. (iii): C2 H22 O11 is the formula for sucrose (sugar). Balance the reaction in which sucrose combines with oxygen to produce carbon dioxide and water. (iv): Balance the chemical reaction in which NO2 and H2 O combine to form HNO2 and HNO3 . (v): Balance the reaction in which the chemical compounds P bN6 and CrMn2 O8 combine to produce P b3 O4 , CrO3, MnO2 and NO.

The subject at the head-end of each arrow is an extension of the one at the tail-end. 1: Data on the thickness of various US coins is given below. 00 Formulate a version of the coin assembly problem in Example 3 in 26 Ch. 1. Linear eqs. 35 mm, as a system of linear equations by ignoring the nonnegative integer requirements on the decision variables. 2: In the year 2000, the US government introduced two new one dollar coins. Data on them is given below. 1 above. 1. 1; in addition, the set is required to contain exactly two Sacagawea golden dollars, and exactly 3 Sacagawea silver toned dollars.

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